We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the Variational Multiscale Method in the context of a space-time formulation and computes a coarse-scale representation of the differential operator that is enriched by auxiliary space-time corrector functions. Once computed, the coarse-scale representation allows us to efficiently obtain well-approximating discrete solutions for multiple right-hand sides. We prove first-order convergence independently of the oscillation scales in the coefficient and illustrate how the space-time correctors decay exponentially in both space and time, making it possible to localize the corresponding computations. This localization allows us to define a practical and computationally efficient method in terms of complexity and memory, for which we provide a posteriori error estimates and present numerical examples.

中文翻译：

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抛物线问题的时空多尺度方法

我们提出了一种抛物线模型问题的时空多尺度方法，其潜在系数可能在空间和时间变量方面都具有高度振荡性。该方法基于时空公式上下文中的变分多尺度方法的框架，并计算微分算子的粗尺度表示，该表示由辅助时空校正函数丰富。一旦计算，粗尺度表示允许我们有效地获得多个右侧的近似离散解。我们证明了独立于系数中的振荡尺度的一阶收敛，并说明了时空校正器如何在空间和时间上呈指数衰减，从而可以定位相应的计算。